On the other hand, it is unknown whether \(\chi _P(G)\)  can be bounded above by a linear or even polynomial function of \(\chi _{\ell }(G)\) , and it is unknown whether \(\chi _{DPP}(G)\)  can be bounded above by a linear function of \(\chi _{DP}(G)\) . Duraj, Gutowski, and Kozik [1]} have pointed out that currently, the best known bound for \(\chi _P(G)\)  in terms of \(\chi _{\ell }(G)\)  comes from the inequality chain
\(\chi _P(G) \le d(G)+1 \le 4 {{s^4 } \atopwithdelims ()s} \log \left( 2{{s^4 } \atopwithdelims ()s} \right) + 1,\) 
